The Fredholm Index for Elements of Toeplitz-Composition C<Superscript>*</Superscript>-Algebras

We analyze the essential sectrum and index theory of elements of Toeplitz-composition C^*-algebras (algebras generated by the Toeplitz algebra and a single linear-fractional composition operator, acting on the Hardy space of the unit disk). For automorphic composition operators we show that the quotient of the Toeplitz-composition algebra by the compacts is isomorphic to the crossed product C^*-algebra for the action of the symbol on the boundary circle. Using this result we obtain sufficient conditions for polynomial elements of the algebra to be Fredholm, by analyzing the spectrum of elements of the crossed product. We also obtain an integral formula for the Fredholm index in terms of a generalized Chern character. Finally we prove an index formula for the case of the non-parabolic, non-automorphic linear fractional maps studied by Kriete, MacCluer and Moorhouse.     

Zeno Product Formula Revisited

We introduce a new product formula which combines an orthogonal projection with a complex function of a non-negative operator. Under certain assumptions on the complex function the strong convergence of the product formula is shown. Under more restrictive assumptions even operator-norm convergence is verified. The mentioned formula can be used to describe Zeno dynamics in the situation when the usual non-decay measurement is replaced by a particular generalized observables in the sense of Davies.     

One more proof of the Borodin-Okounkov formula for Toeplitz determinants

Recently, Borodin and Okounkov [2] established a remarkable identity for Toeplitz determinants. Two other proofs of this identity were subsequently found by Basor and Widom [1], who also extended the formula to the block case. We here give one more proof, also for the block case. This proof is based on a formula for the inverse of a finite block Toeplitz matrix obtained in the late seventies by Silbermann and the author.     

Index formulas for countably k-set contractive operators

We give an index formula for countably k$k$-set contractive bounded linear operators in a real Banach space, by using a degree theory for countably condensing operators. As a consequence, we present another index formula for differentiable countably k$k$-set contractive operators.     

Products of Toeplitz Operators on the Bergman Space

In 1962 Brown and Halmos gave simple conditions for the product of two Toeplitz operators on Hardy space to be equal to a Toeplitz operator. Recently, Ahern and Cucković showed that a similar result holds for Toeplitz operators with bounded harmonic symbols on Bergman space. For general symbols, the situation is much more complicated. We give necessary and sufficient conditions for the product to be a Toeplitz operator (Theorem 6.1), an explicit formula for the symbol of the product in certain cases (Theorem 6.4), and then show that almost anything can happen (Theorem 6.7).     

Paraproducts and Hankel operators of Schatten class via p-John-Nirenberg Theorem

We give an interpolation-free proof of the known fact that a dyadic paraproduct is of Schatten-von Neumann class S_p, if and only if its symbol is in the dyadic Besov space B_p^d. Our main tools are a product formula for paraproducts and a “p-John-Nirenberg-Theorem” due to Rochberg and Semmes.We use the same technique to prove a corresponding result for dyadic paraproducts with operator symbols.Using an averaging technique by Petermichl, we retrieve Peller's characterizations of scalar and vector Hankel operators of Schatten-von Neumann class S_p for 1<p<∞. We then employ vector techniques to characterise little Hankel operators of Schatten-von Neumann class, answering a question by Bonami and Peloso.Furthermore, using a bilinear version of our product formula, we obtain characterizations for boundedness, compactness and Schatten class membership of products of dyadic paraproducts.     

Scattering Matrix,Phase Shift,Spectral Shift and Trace Formula for One-dimensional Dissipative Schrödinger-type Operators

The paper is devoted to Schr?dinger operators with dissipative boundary conditions on bounded intervals. In the framework of the Lax-Phillips scattering theory the asymptotic behaviour of the phase shift is investigated in detail and its relation to the spectral shift is discussed. In particular, the trace formula and the Birman-Krein formula are verified directly. The results are exploited for dissipative Schr?dinger-Poisson systems. In friendship dedicated to P. Exner on the occasion of his 60^th birthday This work was supported by DFG, Grant 1480/2.     

The contraction rate in Thompson's part metric of order-preserving flows on a cone – Application to generalized Riccati equations

We give a formula for the Lipschitz constant in Thompson's part metric of any order-preserving flow on the interior of a (possibly infinite dimensional) closed convex pointed cone. This shows that in the special case of order-preserving flows, a general characterization of the contraction rate in Thompson's part metric, given by Nussbaum, leads to an explicit formula. As an application, we show that the flow of the generalized Riccati equation arising in stochastic linear quadratic control is a local contraction on the cone of positive definite matrices and characterize its Lipschitz constant by a matrix inequality. We also show that the same flow is no longer a contraction in other invariant Finsler metrics on this cone, including the standard invariant Riemannian metric. This is motivated by a series of contraction properties concerning the standard Riccati equation, established by Bougerol, Liverani, Wojtkowski, Lawson, Lee and Lim: we show that some of these properties do, and that some other do not, carry over to the generalized Riccati equation.     

Regularization and error estimate for the nonlinear backward heat problem using a method of integral equation

We consider the inverse time problem for the nonlinear heat equation in the form

     

(Modified) Fredholm Determinants for Operators with Matrix-Valued Semi-Separable Integral Kernels Revisited

We revisit the computation of (2-modified) Fredholm determinants for operators with matrix-valued semi-separable integral kernels. The latter occur, for instance, in the form of Greens functions associated with closed ordinary differential operators on arbitrary intervals on the real line. Our approach determines the (2-modified) Fredholm determinants in terms of solutions of closely associated Volterra integral equations, and as a result offers a natural way to compute such determinants.We illustrate our approach by identifying classical objects such as the Jost function for half-line Schrödinger operators and the inverse transmission coe.cient for Schrödinger operators on the real line as Fredholm determinants, and rederiving the well-known expressions for them in due course. We also apply our formalism to Floquet theory of Schrödinger operators, and upon identifying the connection between the Floquet discriminant and underlying Fredholm determinants, we derive new representations of the Floquet discriminant.Finally, we rederive the explicit formula for the 2-modified Fredholm determinant corresponding to a convolution integral operator, whose kernel is associated with a symbol given by a rational function, in a straghtforward manner. This determinant formula represents a Wiener-Hopf analog of Days formula for the determinant associated with finite Toeplitz matrices generated by the Laurent expansion of a rational function.     

Kernels of Hankel operators and hyponormality of Toeplitz operators

We give a formula for and describe when . We explore the hyponormality of Toeplitz operators whose symbols are of circulant type and some more general types. In addition, we discuss formulas for and estimates of the rank of the self-commutator of a hyponormal Toeplitz operator. Received September 17, 1999 / Revised May 25, 2000 / Published online December 8, 2000     

Invariant Subspaces of Collectively Compact Sets of Linear Operators

In this paper, we first give some invariant subspace results for collectively compact sets of operators in connection with the joint spectral radius of these sets. We then prove that any collectively compact set M in algΓ satisfies Berger-Wang formula, where Γ is a complete chain of subspaces of X.      

Behaviour of the approximation numbers of a Sobolev embedding in the one-dimensional case

We consider the Sobolev embeddings

     

On the Determinant of a Certain Wiener-Hopf + Hankel Operator

We establish an asymptotic formula for determinants of truncated Wiener-Hopf+Hankel operators with symbol equal to the exponential of a constant times the characteristic function of an interval. This is done by reducing it to the corresponding (known) asymptotics for truncated Toeplitz+Hankel operators. The determinants in question arise in random matrix theory in determining the limiting distribution for the number of eigenvalues in an interval for a scaled Laguerre ensemble of positive Hermitian matrices.     

Realization of the Stasheff polytope

We propose a simple formula for the coordinates of the vertices of the Stasheff polytope (alias associahedron) and we compare it to the permutohedron.     

The higher-order derivatives of spectral functions

We are interested in higher-order derivatives of functions of the eigenvalues of real symmetric matrices with respect to the matrix argument. We describe a formula for the k-th derivative of such functions in two general cases.The first case concerns the derivatives of the composition of an arbitrary (not necessarily symmetric) k-times differentiable function with the eigenvalues of symmetric matrices at a symmetric matrix with distinct eigenvalues.The second case describes the derivatives of the composition of a k-times differentiable separable symmetric function with the eigenvalues of symmetric matrices at an arbitrary symmetric matrix. We show that the formula significantly simplifies when the separable symmetric function is k-times continuously differentiable.As an application of the developed techniques, we re-derive the formula for the Hessian of a general spectral function at an arbitrary symmetric matrix. The new tools lead to a shorter, cleaner derivation than the original one.To make the exposition as self contained as possible, we have included the necessary background results and definitions. Proofs of the intermediate technical results are collected in the appendices.     

(Modified) Fredholm Determinants for Operators with Matrix-Valued Semi-Separable Integral Kernels Revisited

We revisit the computation of (2-modified) Fredholm determinants for operators with matrix-valued semi-separable integral kernels. The latter occur, for instance, in the form of Greens functions associated with closed ordinary differential operators on arbitrary intervals on the real line. Our approach determines the (2-modified) Fredholm determinants in terms of solutions of closely associated Volterra integral equations, and as a result offers a natural way to compute such determinants.We illustrate our approach by identifying classical objects such as the Jost function for half-line Schrödinger operators and the inverse transmission coefficient for Schrödinger operators on the real line as Fredholm determinants, and rederiving the well-known expressions for them in due course. We also apply our formalism to Floquet theory of Schrödinger operators, and upon identifying the connection between the Floquet discriminant and underlying Fredholm determinants, we derive new representations of the Floquet discriminant.Finally, we rederive Böttchers formula for the 2-modified Fredholm determinant corresponding to a convolution integral operator, whose kernel is associated with a symbol given by a rational function, in a straghtforward manner. This determinant formula represents a Wiener-Hopf analog of Days formula for the determinant associated with finite Toeplitz matrices generated by the Laurent expansion of a rational function.     

Kreĭn–Saakyan and Trace Formulas for a Pair of Canonical Differential Operators

An analog of the Kreĭn–Saakyan formula is derived for any pair of relatively prime self-adjoint extensions of a minimal symmetric canonical differential operator. This allows us to deduce a trace formula in the matrix case. I am grateful to Sh. Saakyan for his interest in this work and lively discussion. Received: December 8, 2006. Accepted: December 30, 2006.     

Projections as averages of isometries on minimal norm ideals

In this paper, we consider projections on minimal norm ideals of B(H) that are represented as the average of two surjective isometries. We describe projections of the form